NBER WORKING PAPER SERIES PRODUCTIVITY AND MISALLOCATION IN GENERAL EQUILIBRIUM. David Rezza Baqaee Emmanuel Farhi

Transcription

1 NBER WORKING PAPER SERIES PRODUCTIVITY AND MISALLOCATION IN GENERAL EQUILIBRIUM. David Rezza Baqaee Emmanuel Farhi Working Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA November 2017, Revised March 2018 We thank Philippe Aghion, Pol Antras, Andrew Atkeson, Susanto Basu, John Geanakoplos, Ben Golub, Gita Gopinath, Dale Jorgenson, Marc Melitz, Ben Moll, Matthew Shapiro, Dan Trefler, Venky Venkateswaran, and Jaume Ventura for their valuable comments. We thank German Gutierrez, Thomas Philippon, Jan De Loecker, and Jan Eeckhout for sharing their data. We thank Thomas Brzustowski and Maria Voronina for excellent research assistance. We are especially grateful to Natalie Bau for detailed conversations. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications by David Rezza Baqaee and Emmanuel Farhi. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including notice, is given to the source.

2 Productivity and Misallocation in General Equilibrium. David Rezza Baqaee and Emmanuel Farhi NBER Working Paper No November 2017, Revised March 2018 JEL No. D24,D33,D42,D43,D5,D50,D57,D61,E01,E1,E25,E52,O4,O41 ABSTRACT We provide a general non-parametric formula for aggregating microeconomic shocks in general equilibrium economies with distortions such as taxes, markups, frictions to resource reallocation, and nominal rigidities. We show that the macroeconomic impact of a shock can be boiled down into two components: its pure technology effect; and its effect on allocative efficiency arising from the reallocation of resources, which can be measured via changes in factor income shares. We derive a formula showing how these two components are determined by structural microeconomic parameters such as elasticities of substitution, returns to scale, factor mobility, and network linkages. Overall, our results generalize those of Solow (1957) and Hulten (1978) to economies with distortions. As examples, we pursue some applications focusing on markup distortions. We operationalize our non-parametric results and show that improvements in allocative efficiency account for about 50% of measured TFP growth over the period We implement our structural results and conclude that eliminating markups would raise TFP by about 20%, increasing the economywide cost of monopoly distortions by two orders of magnitude compared to the famous 0.1% estimate by Harberger (1954). David Rezza Baqaee London School of Economics UK D.R.Baqaee@lse.ac.uk Emmanuel Farhi Harvard University Department of Economics Littauer Center Cambridge, MA and NBER emmanuel.farhi@gmail.com A Proofs is available at

3 1 Introduction The foundations of macroeconomics rely on Domar aggregation: changes in a constantreturns-to-scale index are approximated by a sales-weighted average of the changes in its components. 1 Hulten (1978), building on the work of Solow (1957), provided a rationale for using Domar aggregation to interpret the Solow residual as a measure of aggregate TFP. He showed that in efficient economies Y Y L f Λ f L f f i λ i TFP i TFP i, where Y is real GDP, L f is the supply of factor f, Λ f is its of total income share in GDP, TFP i is the TFP of producer i, λ i is its sales as a share of GDP. Although Hulten s theorem is most prominent for its use in growth accounting, where it is employed to measure movements in the economy s production possibility frontier, it is also the benchmark result in the resurgent literature on the macroeconomic impact of microeconomic shocks in mutisector models and models with production networks. 2 The non-parametric power of Hulten s theorem comes from exploiting a macroenvelope condition resulting from the first welfare theorem. This requires perfect competition and Pareto-efficiency. Without these conditions, Hulten s theorem generally fails. 3 Our paper generalizes Hulten s theorem beyond efficient economies, and provides an aggregation result for economies with arbitrary neoclassical production functions, input-output networks, and distortion wedges. Rather than relying on a macro-envelope condition like the first welfare theorem, our results are built on micro-envelope conditions: namely that all producers are cost minimizers. Our result suggests a new and structurally interpretable decomposition of changes in aggregate TFP into pure changes in technology and changes in allocative efficiency. It provides a unified framework for analyzing the effects of distortions and misallocation in general equilibrium economies, the study of which is the subject of a vibrant literature, recently reinvigorated by Restuccia and 1 Although we refer to this idea as Domar aggregation, after Evesy Domar, the basic idea of using sales shares to weight changes in a price or quantity can be traced back at least to the early 18th century writer William Fleetwood. We refer to this idea as Domar aggregation, since Domar (1961) was the first to propose it in the context we are interested in: creating an index of aggregate technical change from measures of microeconomic technical change. 2 See for example Gabaix (2011), Acemoglu et al. (2012), Carvalho and Gabaix (2013), Di Giovanni et al. (2014), Baqaee and Farhi (2017) amongst others. 3 See for example the papers by Basu and Fernald (2002), Jones (2011), Jones (2013), Bigio and La O (2016), Baqaee (2016), or Liu (2017) who explicitly link their inefficient models with the failure of Hulten s result. Some papers which study distorted networked economies (but place less of a focus on how their results compare to Hulten s), are Grassi (2017), Caliendo et al. (2017), Bartelme and Gorodnichenko (2015). 2

4 Rogerson (2008) and Hsieh and Klenow (2009). 4 Loosely speaking, when a producer becomes more productive, the impact on aggregate TFP can be broken down into two components. First, given the initial distribution of resources, the producer increases its output, and this in turn increases the output of its direct and indirect customers; we call this the pure technology effect. Second, the distribution of resources across producers shifts in response to the shock, increasing some producers output and reducing that of others; we call the impact of this reallocation of resources on aggregate TFP the change in allocative efficiency. In efficient economies, changes in allocative efficiency are zero to a first order, and so the overall effect characterized by Hulten (1978) boils down to the pure technology effect. In inefficient economies, changes in allocative efficiency are nonzero in general. Our theoretical contribution is to fully characterize the macroeconomic impact of microeconomic shocks as well as their decomposition into pure technology effects and changes in allocative efficiency in inefficient economies. We present both ex-post and ex-ante results. The ex-post reduced-form results do not require any information about the microeconomic production functions besides inputoutput expenditure shares. The downside of these results is that they depend on the observation of factor income shares before and after the shock. The second set of results are ex-ante structural results. Although they do not necessitate ex-post information, they require information about microeconomic elasticities of substitution. Using this information in conjunction with input-output expenditure shares, we can deduce the implied changes in factor income shares. As a side benefit, our ex-ante results determine how factor income shares respond to shocks for a general neoclassical production structure, which is a question of independent interest in studies of inequality. 5 Our ex-post reduced-form results provide a natural decomposition of aggregate TFP into pure technology effects and changes in allocative efficiency. We compare our decomposition to other decompositions in the growth accounting literature. We focus in particular on the prominent contributions of Basu and Fernald (2002) and Petrin and Levinsohn (2012). We contrast these decompositions with ours and argue why we find ours preferable. 4 Some other prominent examples are Hopenhayn and Rogerson (1993), Banerjee and Duflo (2005), Chari et al. (2007), Guner et al. (2008), Buera et al. (2011), Epifani and Gancia (2011), Fernald and Neiman (2011), Buera and Moll (2012), D Erasmo and Moscoso Boedo (2012), Bartelsman et al. (2013), Caselli and Gennaioli (2013), Oberfield (2013), Peters (2013), Reis (2013), Asker et al. (2014), Hopenhayn (2014), Moll (2014), Midrigan and Xu (2014), Sandleris and Wright (2014), Edmond et al. (2015), Gopinath et al. (2017), and Sraer and Thesmar (2018). 5 See, for example, Piketty (2014), Elsby et al. (2013), Barkai (2016), Rognlie (2016), Koh et al. (2016), and Gutierrez (2017). 3

5 Although we view our main contribution as theoretical, we also demonstrate the empirical relevance and the scope of applicability of our framework via some examples. The examples are proof of concept illustrations of how our results can be used in practice. Specifically, we use our framework to answer three different questions about the role of markups on aggregate productivity. We focus on markups in light of the accumulating evidence that average markups have increased over the past decades in the US How have changes in allocative efficiency contributed to measured TFP growth in the US over the past 20 years? We perform a non-parametric decomposition of measured TFP growth as captured by the Solow residual into a pure technology effect and an allocative efficiency effect. 7 We implement our Solow residual decomposition in the US over the period Focusing on markups as a source of distortions, we find that the improvement in allocative efficiency accounts for about 50% of the cumulated Solow residual. This occurs despite the fact that average markups have been increasing. A rough intuition for this surprising result is that average markups have been increasing primarily due to an across-firms composition effect, whereby firms with high markups have been getting larger, and not a within-firm increase in markups. 8 From a social perspective, these high-markup firms were too small to begin with, and so the reallocation of factors towards them has improved allocative efficiency and TFP. 2. What are the gains from reducing markups in the US, and how have these gains changed over time? Using our structural results, we find that in the US in , eliminating markups would raise aggregate TFP by about 20%. This increases the estimated cost of monopoly distortions by two orders of magnitude compared to the famous estimates of 0.1% of Harberger (1954). 9 6 See Barkai (2016) and Caballero et al. (2017) for arguments using aggregate data, and Gutierrez (2017), and De Loecker and Eeckhout (2017) for evidence using firm-level data. 7 There is also an additional effect, reminiscent of Hall (1990), due to the fact that the Solow residual does not weigh changes in factor shares correctly in the presence of distortions. 8 This is consistent with Vincent and Kehrig (2017) and Autor et al. (2017) who argue that the labor share of income has decreased because more low labor share firms have become larger, and not because the labor share has declined within firms. 9 Harberger s result had a profound impact on the economics discipline by providing an argument for de-emphasizing microeconomic inefficiencies in comparison to Keynesian macroeconomic inefficiencies. This impact is perhaps best illustrated by Tobin s famous quip that it takes a heap of Harberger triangles to fill an Okun gap. 4

6 The reasons for this dramatic difference are that we use firm-level data, whereas Harberger only had access to sectoral data, and that the dispersion of markups is higher across firms within a sector than across sectors. Moreover, the relevant elasticity of substitution is higher in our exercise than in Harberger s since it applies across firms within a sector rather than across sectors. Finally, we properly take into account the general equilibrium input-output structure of the economy to aggregate the numbers in all industries whereas Harberger focused on manufacturing and ignored input-output linkages. Like Harberger, we measure only the static gains from eliminating markups, holding fixed technology, abstracting away from the possibility that lower markups may reduce entry and innovation. In other words, even if markups play an important role in incentivizing entry and innovation, their presence also distorts the allocation of resources, and this latter effect is what we quantify. Interestingly, we also find that the gains from reducing markups have increased since Roughly speaking, this occurs because the dispersion in markups has increased over time. This finding may appear to contradict our conclusion that allocative efficiency has made a positive contribution to measured TFP growth over the period. The resolution is that these results are conceptually different: one is about the contribution of changes in allocative efficiency to measured TFP growth along the observed equilibrium path of the economy, while the other one is about the comparison of the distance from the efficient production possibility frontier at the beginning and at the end of the sample. This distinction highlights the subtleties involved in defining and interpreting different notions of allocative efficiency. 3. How do markups affect the macroeconomic impact and diversification of microeconomic shocks? Our ex-ante structural results allow us to conclude that markups materially affect the impact of microeconomic productivity and markup shocks on output, both at the sector and at the firm level. They amplify some shocks and attenuate others. Unlike a perfectly competitive model, shocks to industries and firms have different effects on output, even controlling for size. Firm-level shocks trigger larger reallocations of resources across producers than industry-level shocks (since firms are more substitutable). On the whole, we find that output is more volatile than in a perfectly competitive model, especially with respect to firm-level shocks. Despite their generality, our results have some important limitations. First, our basic framework accommodates neoclassical production with decreasing or constant returns 5

7 to scale. It can also easily handle fixed costs, as long as production has constant or increasing marginal cost. However, it is unable to deal with non-neoclassical production featuring increasing returns such as those studied by Baqaee (2016), where by increasing returns, we refer to a situation where marginal variable costs are decreasing in output. In the appendix, we sketch how our results can be extended to cover such cases, when we discuss entry and exit. Second, in this paper we focus on first-order approximations. We show that under some conditions, the nonlinear analysis of efficient economies in Baqaee and Farhi (2017) can be leveraged to characterize nonlinearities in the sort of inefficient economies studied in this paper. Finally, we model frictions using wedges, which we take as primitives. The advantage is that we characterize the response of the decentralized equilibrium to a change in the wedges without committing to any particular theory of wedge determination. The downside is that this makes it hard to perform counterfactuals when wedges are endogenous. However, in these cases, our results are still relevant as part of a larger analysis that accounts for the endogenous response of wedges. As an example, in the appendix, we show how to use our results to analyze the effect of monetary policy and productivity shocks in a model with sticky prices. The outline of the paper is as follows. In Section 2, we set up the general model, and we prove our non-parametric results. We also discuss how to interpret these results, and the data required to implement our formulas. In Section 3, we introduce a parametric version of the general model and present our structural results. We use a model with CES production and consumption functions, with an arbitrary number of nests, inputoutput patterns, returns to scale, and factors of production. In Section 4 we discuss some subtleties in implementing and interpreting our results, including how to deal with endogenous wedges. In Section 5, we apply our results to the data by performing nonparametric ex-post decompositions of the sources of growth in the US, as well as structural exercises measuring the TFP gains from markup reductions, aggregate volatility arising from micro shocks, and the macro impact of micro shocks, in a calibrated model. In Section 6, we describe extensions of our results that account for endogenous factor supply, fixed costs, entry, and nonlinearities. 2 General Framework and Non-Parametric Results We set up our general framework, and characterize how shocks to wedges and productivity affect equilibrium output and TFP. We define our notion of change in allocative efficiency. We explain how it leads to a new decomposition of the Solow residual into changes in pure technology and changes in allocative efficiency. We end by discussing 6

8 the relationship between our results and the rest of the literature. 2.1 Set up The model has N producers indexed by i and F inelastic factors indexed by f with supply L f. The output of each producer is produced using intermediate inputs and factors, and is sold as an intermediate good to other producers and as a final good. Final Demand Final demand, or GDP, in the economy is represented as the maximization of a constantreturns aggregator of final demand for individual goods subject to the budget constraint Y = max {c 1,...,c N } D(c 1,..., c N ) N (1 + τ 0i )p i c i = i F w f L f + f =1 N π i + τ, i=1 where p i is the price of good i, w f is the wage of factor f, τ 0i is the consumption wedge on good i, π i is the profits of the producer of good i, and τ is a net lump-sum rebate. Producers Good i is produced by producer i according to a constant-returns technology described by the constant-returns cost function 1 A i C i ( (1 + τi1 )p 1,..., (1 + τ in )p N, (1 + τ f i1 )w 1,..., (1 + τ f if )w F) yi, where A i is a Hicks-neutral productivity shifter, y i is total output, τ ij is the input-specific tax wedge on good j, and τ f is a factor-specific tax wedge on factor g. We assume that ig producer i sets a price p i = µ i C i /A i equal to an exogenous markup µ i over marginal cost C i /A i. General Equilibrium Given productivities A i, markups µ i, wedges τ ij and τ f, general equilibrium is a set of ij prices p i, factor wages w f, intermediate input choices x ij, factor input choices l i f, outputs y i, 7

9 and final demands c i, such that: each producer minimizes its costs and charges the relevant markup on its marginal cost; final demand maximizes the final demand aggregator subject to the budget constraint, where profits and revenues from wedges are rebated lump sum; and the markets for all goods and factors clear. Two Simplifications: Constant Returns to Scale and Markup-Wedge Equivalence Without loss of generality, we exploit two simplifications. First, despite specifying constant-returns cost functions, our setup can accommodate decreasing returns to scale. This is because decreasing returns to scale can be modeled with constant returns to scale and producer-specific fixed factors. Going forward, we proceed with our constant-returns setup with the understanding that it can be reinterpreted to capture decreasing returns provided that the original set of factor is expanded to include producer-specific fixed factors. Second all the wedges τ ij and τ f can be represented as markups in a setup with ig additional producers. For example, the good-specific wedge τ ij in the original setup can be modeled in a modified setup as a markup charged by a new producer which buys input j and sells it to producer i. Going forward, we take advantage of this equivalence and assume that all wedges take the form of markups. 2.2 Input-Output Definitions To state our generalization of Hulten s theorem, we introduce some input-output notation and definitions. Our results are comparative statics describing how, starting from an initial decentralized equilibrium, the equilibrium level of output changes in responses to shocks to productivities A k and markups µ k. Without loss of generality, we normalize the initial productivity levels to one. We now define input-output objects such as input-output matrices, Leontief inverse matrices, and Domar weights. Each of these quantities has a revenue-based version and a cost-based version, and we present both. All these objects are defined at the initial equilibrium. Final Expenditure Shares Let b be the N 1 vector whose ith element is equal to the share of good i in final expenditures b i = p i c i N j=1 p j c j, 8

10 where the sum of final expenditures N j=1 p j c j is nominal GDP. Input-Output Matrices To streamline the exposition, we treat factors as special endowment producers which do not use any input to produce. We form an (N + F) 1 vector of producers, where the first N elements correspond to the original producers and the last F elements to the factors. For each factor, we interchangeably use the notation w f or p N+ f to denote its wage, and the notation L i f or x i(n+ f ) to denote its use by producer i. We define the revenue-based input-output matrix Ω to be the (N + F) (N + F) matrix whose ijth element is equal to i s expenditures on inputs from j as a share of its total revenues Ω ij p jx ij p i y i. The first N rows and columns of Ω correspond to goods, and the last F rows and columns correspond to the factors of production. Since factors require no inputs, the last F rows of Ω are identically zero. Similarly, we define the cost-based input-output matrix Ω to be the (N + F) (N + F) matrix whose ijth element is equal to the elasticity of i s marginal costs relative to the price of j Ω ij log C i log p j = p j x ij N+ f k=1 p kx ik. The second equality uses Shephard s lemma to equate the elasticity of the cost of i to the price of j to the expenditure share of i on j. Since factors require no inputs, the last F rows of Ω are identically zero. The revenue-based and cost-based input-output matrices are related according to Ω = diag(µ)ω where µ is the vector of markups/wedges, and diag(µ) is the diagonal matrix with ith diagonal element given by µ i. Leontief Inverse Matrices We define the revenue-based and cost-based Leontief inverse matrices as Ψ (I Ω) 1 = I + Ω + Ω and Ψ (I Ω) 1 = I + Ω + Ω

11 While the input-output matrices Ω and Ω record the direct exposures of one producer to another, in revenues and in costs respectively, the Leontief inverse matrices Ψ and Ψ record instead the direct and indirect exposures through the production network. This can be seen most clearly by noting that (Ω n ) ij and ( Ω n ) ij measure the weighted sums of all paths of length n from producer i to producer j. Domar Weights We define the revenue-based Domar weight λ i of producer i to be its sales share as a fraction of GDP λ i p i y i N j=1 p j c j. Note that N i=1 λ i > 1 in general since some sales are not final sales but intermediate sales. The accounting identity N p i y i = p i c i + p i x ji = b i p j c j + Ω ji p j y j relates Domar weights to the Leontief inverse via j j=1 j λ = b Ψ = b I + b Ω + b Ω (1) Similarly, we define the vector of cost-based Domar weights to be λ b Ψ = b I + b Ω + b Ω We choose the name cost-based Domar weight for λ to contrast it with the traditional revenue-based Domar weight λ. Intuitively, λ k measures the importance of k as a supplier in final demand, both directly and indirectly through the network. 10 This can be seen most clearly by noting that the i-th element of b Ω n measures the weighted sum of all paths of length n from producer i to final demand. For expositional convenience, for a factor f we use Λ f and Λ f instead of λ f and λ f. Note that revenue-based Domar weight Λ f of factor f is simply its income share. 10 The cost-based Domar weight only depends on k s role as a supplier rather than its role as a consumer. It is also sometimes referred to as the influence vector, since in a certain class of models (like Jones, 2013; Acemoglu et al., 2012), it maps micro productivity shocks to output. We avoid this language since influence is an ambiguous term, and while the cost-based Domar weights are often-times useful in characterizing equilibria, they do not generally map productivity shocks to output. In other words, they are not generally equivalent to influence. 10

12 Cross-Entropy Our final input-output definition is the cross-entropy, which loosely speaking, is a measure of difference between distributions. 11 The cross-entropy between Λ and Λ is ( ) F H( Λ, Λ) E Λ log Λ = f =1 Λ f log Λ f. Here Λ and Λ are seen as measures on the set of factors. Since f Λ f = 1, the total mass of Λ is equal to one and hence it can be interpreted as a probability distribution. By contrast, Λ is typically not a probability distribution since f Λ f 1 in general. However, we can always supplement Λ with the pure profit share Λ f = 1 f Λ f accruing to an extra factor f for which the cost-based share is zero Λ f interpretation. = 0, and hence recover a probabilistic For a given change d log Λ in the probability distribution for Λ resulting from a combination of productivity shocks d log A and wedge shocks d log µ. We denote by d H( Λ, Λ) H( Λ, Λ + d Λ) H( Λ, Λ) = F f =1 Λ f d log Λ f, the change in relative entropy of Λ with respect to the fixed initial distribution Λ. 2.3 Comparative-Static Results In this section, we derive our comparative-static results. Take as given the factor supplies L f, the cost functions C i, and final Demand D. Let X be an (N + F) (N + F) admissible allocation matrix, where X ij = x ij /y j is the share of the physical output y j of producer j used by producer i. Specify the vector of productivities A and denote by Y(A, X) the output Y achieved by this allocation. 12,13 Finally, define X ij (A, µ) to be equal to x ij (A, µ)/y j (A, µ) 11 Cross-entropy between two distributions is minimized when the two distributions are the same. This definition is due to Claude Shannon (1948). Note that the Kullback and Leibler (1951) divergence is crossentropy plus a constant, so that d H( Λ, Λ) = d D KL ( Λ Λ) in our context. Formally, this divergence is not a distance function, but a premetric. 12 The allocation matrix is admissible if the following conditions are verified: 0 X ij 1 for all i and j; X ij = 0 for all j and for N + 1 i N + F; N+F i=1 X ij 1 for all 1 j N; N+F i=1 X ij = 1 for all N + 1 j N + F; and there exists a unique resource-feasible allocation such that the share x ij /y j of the output y j of producer i which is used by producer j is equal to X ij, so that X ij = x ij y j. 13 To see how to construct this allocation, consider the production functions F i defined as the duals of the cost functions C i in the usual way. Then the vector of outputs y i solves the system of equations y i = F i (X 1i y 1,..., X (N+F)i y N+F ) for 1 i N and y N+ f = L f for 1 f F. The corresponding level of final 11

13 at the decentralized equilibrium when the vector of productivities is A and the vector of wedges is µ. The level of output at this equilibrium is given by Y(A, X(A, µ)). Now consider how the general equilibrium level of output changes in response to shocks d log A and d log µ: d log Y = log Y log A d log A + log Y X d X } {{ }} {{ } Technology Allocative Efficiency The change in output can be broken down into two components: the direct or pure effect of changes in technology d log A, holding the distribution of resources X constant; and the indirect effects arising from the equilibrium changes in the distribution of resources d X. Essentially, changes in allocative efficiency are the gap that opens up following a shock between the equilibrium level of output and a passive allocation that just scales the initial allocation proportionately without allowing any other form of reallocation through substitution. The passive allocation constitutes a benchmark without reallocation, and so it stands a useful yardstick against which to measure changes in allocative efficiency in the equilibrium allocation. 14 Now, we extend Hulten (1978) to cover inefficient economies and provide an interpretation for the result. We also extend Hulten s theorem along another dimension by characterizing changes in output following changes in wedges. Theorem 1. Consider some distribution of resources X corresponding to the general equilibrium allocation at the point (A, µ), then where and d log Y = log Y log Y d log A + log A X d X log Y log A d log A = λ d log A log Y X d X = λ d log µ + d H( Λ, Λ) = λ d log µ Λ d log Λ. consumption of good i is c i = y i (1 N+F j=1 X ji) and the level of output is D(c 1,..., c N ). 14 The changes in the passive allocation in response to productivity shocks d log A are easily derived. Since the elasticity of a production function to an input is given by its cost share, we have log Y/ log A i = N j=1 b j log y j / log A i, where y j / A i must solve the following system of equations: log y j / log A i = N k=1 Ω jk log y k / log A i + δ ji. This implies that log y j / log A i = b j Ψ ji and log Y/ log A = Ψ b = λ. Moreover, the passive allocation is invariant to wedge shocks so that log y j / log µ i =

14 This implies that d log Y d log A k = λ k + d H( Λ, Λ) d log A k = λ k f Λ f d log Λ f d log A k, (2) and d log Y d log µ k = λ k + d H( Λ, Λ) d log µ k = λ k f Λ f d log Λ f d log µ k. (3) Theorem 1 not only provides a formula for the macroeconomic output impact of microeconomic productivity and wedges shocks, but it also provides an interpretable decomposition of the effect. Specifically, the first component ( log Y/ log A) d log A = λ d log A is the pure technology effect: the change in output holding fixed the share of resources going to each user; the second component ( log Y/ X) d X = λ d log µ + d H( Λ, Λ) is the change in output resulting from the reallocation of shares of resources across users. In the proof of Theorem 1 in the appendix, we also provide an explicit characterization of d H( Λ, Λ)/ d log A k and d H( Λ, Λ)/ d log µ k in terms of the microeconomic elasticities of substitutions of the production functions and final demand, the properties of the inputoutput network, and the wedges. We present this characterization in the main body of the paper for the more special parametric version of the model in Section 3. We can obtain Hulten s theorem as a special case of Theorem 1 when there are no wedges. Even this special case is actually a slight generalization of Hulten s theorem since it only requires the initial equilibrium to be efficient, whereas Hulten s theorem applies only to the case where the equilibrium is efficient before and after the shock. Corollary 1 (Hulten 1978). If the initial equilibrium is efficient so that there are no wedges so that µ = 1, then d log Y d log A k = λ k and d log Y d log µ k = 0. In efficient economies, the first-welfare theorem implies that the allocation matrix X(A, µ) maximizes output given resource constraints. The envelope theorem then implies that ( log Y/ X) d X = 0 so that there are no changes in allocative efficiency. Furthermore, because of marginal cost pricing, the direct effect of changes in technology are based on the vector of sales shares or revenue-based Domar weights λ and are given by ( log Y/ log A) d log A = λ d log A. Hence, Hulten s theorem is a macro-envelope theorem of sorts. 13

15 fails. When the initial equilibrium is inefficient so that µ 1, this macro-envelope theorem Intuitively, in equilibrium, from a social perspective, some shares are too large and some shares are too small. to changes in output. Equilibrium changes in shares d X can therefore lead This is precisely what we call a change in allocative efficiency ( log Y/ X) d X = λ d log µ + d H( Λ, Λ), which is nonzero in general. Furthermore, because of wedges between prices and marginal costs, the direct effect of changes in technology are now based on the vector of cost-based Domar weights λ rather than on the vector of revenue-based Domar weights λ and are given by ( log Y/ log A) d log A = λ d log A. In the case of productivity shocks, Theorem 1 implies that changes in allocative efficiency are given by a simple sufficient statistic: the weighted average of the change in factor income shares d H( Λ, Λ) = f Λ f d log Λ f. This remarkable property show that it is not necessary to track how the allocation of every single good is changing across its users. Instead, it suffices to track how factor income shares change. Given that Λ = Λ in efficient equilibria, It may seem surprising that an improvement in allocative efficiency comes together with a movement of Λ away from Λ so that d H( Λ, Λ) > 0. However, the intuition is simple. This happens when f Λ f d log Λ f < 0 so that the weighted average of factor shares decreases. This means that the more monopolized or downwardly distorted parts of the economy are receiving more resources. This improves allocative efficiency, since from a social perspective, these monopolized or downwardly distorted parts of the economy receive too few resources to begin with. Similarly, in the case of markup shocks, Theorem 1 implies that changes in allocative efficiency are given by a simple sufficient statistic: λ d log µ + d H( Λ, Λ) = λ d log µ f Λ f d log Λ f. Now d H( Λ, Λ) = f Λ f d log Λ f reflects both the direct effect λ d log µ of the markup change on the profit share and the reallocation of workers towards or away from more distorted producers. To isolate the changes in allocative efficiency, which arise from the latter, we must net out the former. 2.4 Illustrative Examples In this section, we introduce some bare-bones examples to illustrate the intuition of Theorem 1. In Section 3, we specialize Theorem 1 to the case of general nested constantelasticity-of-substitution (CES) economies with arbitrary input-output linkages. The examples that we present here are simple special cases of these more general results. Consider the three economies depicted in Figure 1. In all three economies, there is a single factor called labor. The only distortions in these examples are the markups charged 14

16 L L L N 1 N 1 1 HH HH HH (a) Vertical Economy (b) Horizontal Economy (c) Round-about Economy Figure 1: The solid arrows represent the flow of goods. The flow of profits and wages from firms to households has been suppressed in the diagram. The sole factor for this economy is indexed by L. by the producers. The vertical economy in Figure 1a, drawn from Baqaee (2016), is a chain of producers. Producer N produces linearly using labor and downstream producers transform linearly the output of the producer immediately upstream from them. The household purchases the output of the most downstream producer. The horizontal economy in Figure 1b features downstream producers who produce linearly from labor. The household purchases the output of the downstream producers according to a CES aggregator with elasticity θ 0. The roundabout economy in Figure 1c features only one producer, who combines labor and its own products using a CES production function. These different economies help illustrate the disappearance of two serendipities implied by the assumptions of Hulten s theorem: (1) the equality of revenue-based and cost-based Domar weights (used to weigh the direct or pure effects of technology); and (2) the absence of changes in allocative efficiency (reflecting the efficiency of the initial allocation). The vertical economy breaks (1) but not (2), the horizontal economy breaks (2) but not (1), and the round-about economy breaks (1) and (2). Vertical Economy First, consider the vertical economy in Figure 1a. In this economy, there is only one feasible allocation of resources, so the equilibrium allocation is efficient. An application of Theorem 1 then confirms that d log Y d log A k = λ k Λ L d log Λ L d log A k = λ k d log Λ L d log A k = λ k = 1 15

17 and d log Y d log µ k = λ k Λ L d log Λ L d log µ k = λ k d log Λ L d log µ k = λ k + λ k = 0. Indeed, in this vertical economy, the verification of the theorem is trivial since λ k = 1, Λ L = 1, d log Λ L / d log A k = 0, and d log Λ L / d log µ k = 1. Hence, Hulten s theorem fails in the vertical economy, even though the equilibrium is efficient. Reassuringly, our decomposition detects no changes in allocative efficiency since λ d log µ Λ L d log Λ L = 0. The failure of Hulten s theorem is instead due to the gap between the revenue-based Domar weights λ k = k 1 i=1 µ 1 and the cost-based Domar i weights λ k = 1. Indeed, when markups are positive so that µ i > 1 for all i, we have λ k > λ k. This is a consequence of downstream double-marginalization which divorces the revenues earned by a producer from that producer s share in the costs faced by the household. Horizontal Economy Next we consider the horizontal economy represented in Figure 1b. The consumption of the household, or final demand, is given by Y Y = i θ 0 ( ) θ 0 1 θ 0 1 θ ci 0 ω 0i c, i where θ 0 is the elasticity of substitution in consumption, ω 0i are consumption weights, and variables with overlines in the denominator are normalizing constants measured in the same units as the numerator. An application of Theorem 1 then yields d log Y d log Λ L = λ k Λ L = λ k λ k (θ 0 1) d log A k d log A k µ 1 k j λ j µ 1 j 1 (4) and ( d log Y d log Λ L µ = λ k Λ 1 ) k L = λ k θ 0 1. (5) d log µ k d log µ k i λ i µ 1 i In the horizontal economy, λ k = λ k since there is no downstream double-marginalization. 16

18 The direct or pure effects of a technology shock are still given by λ d log A exactly as in Hulten s theorem. However, technology shocks and markup shocks can now trigger nonzero changes in allocative efficiency λ d log µ Λ L d log Λ L. Consider equation (4): the effects of a positive technology shock d log A k to producer k. Holding fixed the share of labor used by each producer, the productivity shock increases the output of producer k. However, the shock also reduces its price, which in turn increases the demand for its output via a substitution effect. Whether workers are reallocated towards or away from producer k depends on whether the increase in demand from the substitution effect is stronger than the increase in supply from the productivity shock. This in turn hinges on the whether θ 0 is greater than or less than 1, i.e. on the direction of the departure from Cobb Douglas. When θ 0 > 1, workers are reallocated towards producer k. When θ 0 < 1, workers are reallocated away from producer k. And when θ 0 = 1, the allocation of workers is unchanged. Whether these reallocation of workers increase or decrease allocative efficiency and output in turn depends of the comparison of the markup of producer k to the (harmonic) average markup ( i λ i µ 1 i ) 1. When θ 0 > 1, workers are reallocated towards producer k. If its markup is larger than the (harmonic) average markup µ k > ( i λ i µ 1 ) 1, then this producer is too small i from a social perspective to begin with. 15 The reallocation of labor towards producer k therefore improves allocative efficiency and increases output. 16 The opposite occurs when the markup of producer k is smaller than the average markup. This effect works in the opposite direction when θ 0 < 1, since in that case, the shock would reallocate workers away from producer k. Of course, in the Cobb-Douglas case when θ 0 = 1, the allocation of labor remains unchanged, and hence there are no changes in allocative efficiency. 17 All of this information is summarized by a simple sufficient statistic: the change in allocative efficiency is exactly the opposite of the change in the labor share Λ L d log Λ L = d log Λ L (since Λ L = 1). The labor share of income decreases (increases), and allocative efficiency improves (worsens), when workers are reallocated to producers that were too small (large) from a social perspective to begin with because they were charging aboveaverage (below-average) markups. With productivity shocks, the benchmark with no changes in allocative efficiency is Cobb Douglas θ 0 = 1. With markup shocks, the benchmark case with no changes in 15 Note that the average markup is simply the inverse of the labor share so that ( i λ i µ 1 ) 1 = 1/Λ i L. 16 When θ 0 > 1 and producer k is significantly more competitive than the average producer µ k < (θ 0 /(θ 0 1) i λ i µ 1 ) 1, then the reduction in allocative efficiency can be so extreme that a positive productivity shock i can actually reduce output. 17 This last property is a more general property of Cobb-Douglas economies which we shall encounter in Section 3: productivity shocks do not lead to any change in allocative efficiency for Cobb-Douglas economies since their allocation matrix does not depend on the level of productivity. 17

19 allocative efficiency is Leontief θ 0 = 0 instead. For a markup shock d log µ k to producer k, when θ 0 = 0, complementarities are extreme and the household chooses to consume a fixed quantity of each good regardless of its price. As a result, the allocation of labor does not change in response to the shock, and there are therefore no associated changes in allocative efficiency. When θ 0 > 0 instead, the price of producer k increases, the demand for its output decreases, and workers are reallocated away from it. Allocative efficiency and output decrease (increase) if its markup is larger (smaller) than the average markup. All of this information is again summarized by a simple sufficient statistic λ d log µ Λ L d log Λ L = λ d log µ d log Λ L. Now the opposite of the change in the labor share d log Λ L reflects both the direct effect λ d log µ of the markup change on the profit share and the reallocation of workers towards or away from more distorted producers. isolate the changes in allocative efficiency, which arise from the latter, we must net out the former. To Round-about Economy Finally, we consider the round-about economy in Figure 1c. There is a single producer producing using labor and its own goods according to y 1 = A 1 y 1 ω 11 ( x11 x 11 ) θ 0 1 θ 0 + ω 1L ( L1 L 1 ) θ 0 1 θ 0 θ 0 θ 0 1. and An application of Theorem 1 then yields d log Y d log A 1 = λ 1 Λ L d log Λ L d log A 1 = λ 1 (θ 0 1)λ 1 ( λ 1 1)(µ 1 1) d log Y d log µ 1 = λ 1 Λ L d log Λ L d log µ 1 = θ 0 λ 1 ( λ 1 1)(µ 1 1). The round-about economy combines features of the vertical economy and of the horizontal economy. As in the vertical economy, revenue-based and cost-based Domar weights differ since λ 1 = µ 1 /[µ 1 (1 λ 1)] λ 1 1 as long as µ 1 1. As in the horizontal economy, we have non-trivial changes in allocative efficiency in general so that λ d log µ Λ L d log Λ 0. The intuitions for these results combine those of the vertical economy and of the horizontal economy. 18

20 2.5 Growth Accounting In this section, we discuss how to decompose time-series changes in aggregate TFP and the Solow residual into pure technology changes and allocation efficiency changes. For the purpose of this section, we introduce a small but simple modification to allow for changes in factor supplies. We denote the supply of factor f by L f and by L the vector of factor supplies. The impact of a shock to the supply of a factor is given by d log Y/ d log L f = Λ f + d H( Λ, Λ)/ d log L f = Λ f g Λ g d log Λ g / d log L f. Proposition 1 (TFP Decomposition). To the first order, we can decompose aggregate TFP as log Y t Λ t 1 log L t } {{ } Aggregate TFP λ t 1 log A t λ t 1 } {{ } log µ t Λ t 1 log Λ t. (6) } {{ } Technology Allocative Efficiency The left-hand side of this expression, which we define to be aggregate TFP growth, differs from the Solow residual since it weighs the change in L f,t by the cost-based Domar weight Λ f,t rather than the revenue-based Domar weight Λ f,t. The traditional Solow residual attributes all non-labor income to capital (and has no room for profit income). Therefore, with only labor (L) and capital (K) as factors, the traditional Solow residual would be log Y t ˆΛ t 1 log L t λ t 1 log A t λ t 1 log µ t Λ t 1 log Λ t+( Λ t 1 ˆΛ t 1 ) log L t, (7) where ˆΛ L,t 1 = Λ L,t 1 for labor and ˆΛ K,t 1 = 1 Λ L,t 1 for capital. The key difference is that capital is weighed according to 1 Λ L,t 1 and not to Λ K,t 1. Using Λ to weigh factors in (6) is consistent with Hall (1990), who showed that for an aggregate production function, aggregate TFP should weigh changes in factor inputs by their share of total cost rather than their share of total revenue. In our context unlike in Hall s, the equilibrium can be distorted given factor supplies and there is no structural aggregate production function. We must weigh factors by their cost-based Domar weights. Proposition 1 therefore unifies the approach of Hulten (1978), who eschews aggregate production functions, but maintains efficiency, with that of Hall (1990) who does not require efficiency but maintains aggregate production functions and therefore ignores the allocative efficiency issues that most concern us. Turning to the right-hand side, in the case of an efficient economy, the envelope theorem implies that the reallocation terms are welfare-neutral (to a first order) and can be ignored. Furthermore, the appropriate weights on the technology shocks λ t coincide with the observable sales shares. In the presence of distortions, these serendipities disappear. 19

21 However, given the input-output expenditure shares across producers, the level of wedges and their changes, and the changes in factor income shares, we can compute the righthand side of equation (6) without having to make any parametric assumptions. This is an ex-post decomposition in the sense that it requires us to observe factor income shares and factor supplies at the beginning and at the end of the period. It is important to note that Proposition 1 can be used in contexts where productivity or wedges are endogenous to some more primitive fundamental shocks, if these endogenous changes are actually observed in the data. 2.6 Comparison with Basu-Fernald and Petrin-Levinsohn In their seminal work, Basu and Fernald (2002) provide an alternative decomposition of aggregate TFP changes into pure technology changes and changes in allocative efficiency for economies with markups. Their pure technology term, like ours, is a weighted average of technology changes log A kt for each producer. The weight λ kt (1 s Mkt )/(1 µ kt s Mkt ) attached to a given producer k is just its share in value added λ kt (1 s Mkt ) multiplied by a correction 1/(1 µ kt s Mkt ) involving its intermediate input share in revenues s Mkt and its markup µ kt. These weights therefore differ from the cost-based Domar weights λ kt prescribed by our decomposition. In fact, the information required to calculate their weights the value added share, intermediate-input share, and markup of each producer is not enough in general to calculate the cost-based Domar weights. Computing cost-based Domar weights requires the whole input-output matrix, and not just the intermediate input shares. As a result, their decomposition is different from ours. Similar comments apply to Petrin and Levinsohn (2012), who build on the Basu and Fernald (2002) approach with extensions to cover non-neoclassical features of production like entry and fixed or sunk costs. The pure technology changes of Petrin and Levinsohn weigh the underlying technology changes log A kt of all producers by the usual revenuebased Domar weights λ kt, and define changes in allocative efficiency in terms of the gap between the Solow residual and the Domar-weighted growth in productivity. Although we abstract away from fixed costs and entry in our benchmark model, we discuss how the results can be extended to incorporate these features of the data in Appendix C. We demonstrate the difference between our approach and the Basu-Fernald and Petrin- Levinsohn decompositions by way of a revealing example. Consider an economy where the production network Ω is an acylic graph, as illustrated in Figure 2. The term acyclic here means that any two goods are connected to one another by exactly one undirected path, so that each factor and each good has a unique consumer. Such economies have 20

22 a unique resource feasible allocation, simply because there is no option to allocate a given factor or good to different uses. This allocation is necessarily efficient. Markups and wedges have no effect on the allocation of resources, and as a result, there is no misallocation. In other words, misallocation requires cycles (undirected paths that connect a node back to itself) in the production network. F 1 F K 1 N HH Figure 2: An acyclic economy, where the solid arrows represent the flow of goods. The factors are the green nodes. Each supplier (including factors) have at most one customer, whereas a single customer may have more than one supplier. Economies without cycles can be represented as directed trees with the household being the root. Corollary 2 (Acyclic Economies). If the production network of the economy is an acyclic graph, then d log Y d log A k = λ k, d log Y d log µ k = 0. (8) This follows immediately from the fact that by construction, acyclic economies hold fixed the share of resources across producers. An important consequence of this proposition is that for acyclic economies, where it is unambiguous a priori that there is no misallocation, our definition of changes in allocative efficiency indeed identifies that there are no changes in allocative efficiency in response to shocks. On the other hand, the Basu- Fernald and Petrin-Levinsohn decompositions do detect changes in allocative efficiency despite the fact that these economies are efficient. Note that although the equilibrium allocation in this economy is efficient, Hulten s theorem still fails because the observed sales shares λ do not coincide with λ. 18,19 18 See Appendix F for a fully worked-out example. 19 Relative to Basu-Fernald and Petrin-Levinsohn, our approach also economizes on information by recognizing that the system of first-order conditions arising from cost-minimization by every producer gives rise to a system that can be solved. This is what allows us to summarize all the information into changes in the wedges, and changes in the primary factors, whereas the other decompositions require information on tracking how every input is used by every producer. 21